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It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not homeomorphic? (By isospectral, I mean that the Laplace-Beltrami operator on functions has the same spectrum on both manifolds.)



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up vote 15 down vote accepted

There's an example due to Doyle and Rossetti ("Tetra and Didi, the cosmic spectral twins"; http://arxiv.org/abs/math.DG/0407422) of 3-manifolds that are isospectral but not even homeomorphic. I don't know if this was the first such example.

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There's older examples, e.g. of Vigneras: ams.org/mathscinet-getitem?mr=584073 –  Ian Agol May 16 '12 at 4:40
Sunada (ams.org/mathscinet-getitem?mr=782558) gave a general framework in which the results of Vigneras fit. –  Jan Jitse Venselaar May 16 '12 at 9:43
Today I learned that you can't hear the topology of a drum. –  Steve Huntsman May 16 '12 at 15:27
Agol, are you sure, that Vigneras constructs isospectral varieties that are not homeomorphic, as opposed as only not isometric? I haven't read the paper in details, but the mathscinet reviews, the introduction of the paper, and the last theorem (theorem 8) says "isometric". –  Joël May 16 '12 at 16:14
@Jan: It follows from an argument of Chen ( ams.org.proxy.lib.umich.edu/mathscinet/search/… ) that Vigneras' examples cannot be constructed from Sunada's method. –  Ben Linowitz Oct 29 at 5:04

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